Matching algorithms with physics: exact sequences of finite element spaces

In finite element lore the term “unstable discretization” is often applied to situations when approximate solutions develop unphysical “wiggles”, or otherwise behave in a strange manner. A textbook example of such wiggles are Galerkin solutions of a scalar hyperbolic equation with discontinuous data. A more subtle example of instability is the “strange” behavior of mixed finite elements for the Stokes problem when velocity and pressure are approximated by a linear-constant pair. In both cases failure of finite element methods to produce an adequate solution stems from discretization defects, i.e., a choice of finite element spaces that is inconsistent with the problem being solved. For instance, oscillations in the Galerkin method are invoked by symmetric discretization of an advection operator, while the linear-constant pair violates the inf-sup stability condition required for wellposedness of saddle-point optimization problems. The presence of these defects in the finite element method will almost surely lead to a disaster and in that sense they represent strong stability threats. However, a finite element discretization may possess flaws which do not lead to an immediate and obvious failure. Examples include minute oscillations in SUPG methods, approximation of solenoidal fields by approximately divergence free vectors, and many other “small” infractions. Because the impact of these defects is, as